Limit as x -> 5 of (2x^2 - 50)/(x

Step by step

The question

Question: find the limit as x → 5 of (2x² − 50)/(x − 5).

lim x → 5 2x² − 50x − 5 =

1 Put the number in

Put 5 in place of x.

The top is 0, and the bottom is 0.

2(5)² − 505 − 5 = 00

0/0 has no value.

It does not tell us the limit.

It tells us to work more.

00: no value yet. Work more!

Careful!

0/0 is not 0, and it is not 1.

Some piece of the top and the bottom is 0 at 5.

We find it.

Careful: not 00 = 0 ✗

2 Factor

When 0/0 comes out, (x − 5) hides in the top and the bottom.

Factor them.

lim x → 5 2x² − 50x − 5 = lim x → 5 2~~(x − 5)~~(x + 5)~~(x − 5)~~

First take out the 2 that is in every term.

x² − 25 is a difference of two squares: x² minus 5 squared.

So it is (x − 5)(x + 5).

2x² − 50 = 2(x² − 25)

a² − b² = (a − b)(a + b), b = 5

3 Cancel

x gets close to 5, but it is never 5.

So x − 5 is not 0.

x ≠ 5, so x − 5 ≠ 0

A number over itself is 1.

So we cross out (x − 5) on the top and the bottom.

Write what is left: 2(x + 5).

= lim x → 5 2(x + 5)

4 Put the number in again

Now x − 5 is gone, so the bottom is not 0.

Put 5 in again.

= 2(5 + 5)

= 2 × 10

So the limit is 20.

= 20

5 Check

Let's check with numbers close to 5.

x = 4.99 gives about 19.98.

x = 4.99: f(x) ≈ 19.98

x = 5.01 gives about 20.02.

Both are close to 20.

Right!

x = 5.01: f(x) ≈ 20.02

Let's look back.

First: put the number in.

Then: factor.

Then: cancel.

Then: put the number in again.

Last: check.

Answer